The 40Ca(t, d0)41Ca reaction near the coulomb barrier

Nuclear Physics A185 (1972) 337-348; Not to be reproduced

by photoprint

@ North-Holland PubIisb~n~ Co., Amsterdam

or microfilm without written permission from the publisher

THE 40Ca(t, do)4*Ca REACTION NEAR THE COULOMB BARRIER L. J. B. GOLDFARB University

and J. A. GONZALEZ

of Manchester, Ml3 9PL England

M. POSNER ~njuersj~~ of Massachusetts - Boston, Boston, Massachusetts 02116 and Brookhaven National Laboratory,

Upton, New York 11973

K. W. JONES Brookhaven National Laboratory,

Upton, New York 11973 f

Received 27 January 1972 Abstract: Angular distributions of the &‘%Za(t,d0)4’Ca reaction have been measured at energies near the Coulomb barrier at 2.0, 2.5, 3.0 and 3.5 MeV. A DWBA analysis of the data gives a value of the spectroscopic factor of 0.80&0.03, in good agreement with the results of 40Ca(d, p#Ca experiments. The sensitivity of our result to variations in the optical-model parameters and bound-state neutron radius is discussed. NUCLEAR REACTIONS 40Ca(t, d,), EIsb = 2.0, 2.5, 3.0, 3.5 MeV; measured a@); EISb = 1.7 to 3.5 MeV; measured ~(160”) 41Ca deduced spectroscopic factor.

E

1

1. Introduction The (t, d) reaction is ill-favoured owing to the difficulties associated with the use of a radioactive triton beam and also simply because the residual nuclear states are just as well reached by means of the (d, p) reaction. There is however, one essential difference between the two reactions. The Q-value for the (t, d) reaction is (B,-6.25) MeV (where B, is the neutron separation energy in the residual nucleus); and this is some 4 MeV less than the corresponding Q-values for the (d, p) reaction. This difference is of particuIar importance for sub-Coulomb processes, which require for significant yields that the Q-value be close to zero ‘); and for many targets, such subCoulomb (t, d) processes lead to low-lying states in the residual nucleus. If, indeed, the energies in both the entrance and exit channels are sufficiently below the corresponding Coulomb-barrier energies, the nuclear distortion problem is largely avoided, and there is the promise of determining nuclear spectroscopic factors with a high degree of certainty. The description of the (tjd) overlap which is associated with these processes is considerably more complicated than that of the overlap which arises in deuteron stripping, and several attempts have been made to estimate this overlap at various levels of sophistication ’ - “). t Work supported in part by the US Atomic Energy Commission. 337 May

1972

338

L. J. B. GOLDFARB

et al.

It is only at sub-Coulombic energies that the (tld) overlap leads to a simple multiplicative factor N in the expression for the differential cross section. The situation is obscured at higher energies by the need to account for finite-range effects “). The extraction of the normalization factor N through a comparison of theory and experiment is prevented by the presence of additional factors which are not well known such as the spectroscopic factor and the square of the normalization constant for the nuclear single-particle state under consideration. To date, the only nucleus that has been studied under Coulomb conditions through the (t, d) reaction has been Pb and the experimental value for N has been estimated as 4.6irO.8 [ref. “)I and 5.60&O.& [ref. 6)]. Again, these values depend on the assun~pt~ons made for the neutron singleparticle states in lead I). Our present work is concerned with the 40Ca(t, d,)“‘Ca ground state transition. The spectroscopic factor associated with this final state is, in fact, well known as a consequence of many (d, p) analyses ‘- 13). Our purpose is that it be recalculated for the (t, d) ground state transition, having taken, for the (tld) overlap, a parameterization “) consistent with a body of information concerning tritons and deuterons. Our motivation is two-fold: (i) to study the adequacy of the DWBA treatment at subCoulombic energies and (ii) to test the estimate of the (tJd) overlap. The attraction of transfer studies in the sub-Coulomb region lies in the certainty that the transfer is localized at distances far from the nuclear surface. The localization is more certain, the larger the Coulomb parameter q (= ZZ’e’ = hu)- ’ where Ze and Z’e are the nuclear charges and v is the relative velocity). This parameter is however = 3 at triton energies near 3 MeV and therefore, for the range of triton energies from 2 to 3 MeV, there is poor localization. One sees this difficulty in an equivalent fashion by noting that the associated wave length (S 9 fm) is large even when compared to the nuclear size. We find, at the lowest energy, that nuclear distortion is evident and that its importance increases with energy (see fig. 4). A proper DWBA treatment therefore requires knowledge of the elastic scattering in the entrance and exit channels. A description of the calculations is given in sect. 3 following an account of the experimental procedure in sect. 2. The overall agreement between theory and experiment is gratifying and the spectroscopic factor with our choice for the {t/d) overlap is found to be M 0.80 for reactions covering the whole energy range. 2. Experimental technique The equipment used for the experiment was quite standard. The triton beam was produced in the BNL 3.5 MeV electrostatic accelerator and used to bombard a target of natural calcium about 370 pg/cm2 thick which had been evaporated on a 20 pg{crn’ carbon foil. The 40Ca(t, d0)41Ca reaction has a cross section small compared to the cross sections for the many other possible reactions from the calcium target itself and carbon. oxygen and other contaminants, so that it was necessary to use particle identification to pick out the deuterons. The yield of elastically scattered tritons was mea-

‘Va(t,

sured with a monitor

counter

do)“‘Ca

fixed at 90” and was used to normalize

forward angles to the back-angle points. The particle identification system consisted

339

REACTION

points taken at

of two silicon surface-barrier

detectors,

a 31 pm AE detector and a 100 pm E detector. A pulse multiplier circuit was used to produce the identification pulses. The performance of the identification system was monitored during the experiment by viewing a two-dimensional display of identifier pulse versus particle energy. One of the major experimental difficulties was the presence of small amounts of Si on the target, presumably from the silicon oil used in the vacuum system. The Q-value for the ‘*Si(t, d,)29Si reaction is 2.220 MeV which is very close to the Q-value of 2.103 MeV for the 40Ca(t, d,)41Ca reaction. Peaks from the two reactions were adequately separated at lab angles from 160” to 110” and at forward angles smaller than 80”. In the intermediate angular range, the peaks overlapped and an uncertainty is introduced in the data which we estimate to be IO-20 y,i or less, except at 80” where peaks totally overlap to give one symmetric peak. The absolute cross-section scale was established by comparison with the elastic scattering of tritons by calcium at 1.765 MeV. The energy loss of the incident triton beam in the target was found by measuring the width of the peak for elastically scattered tritons and for elastically scattered oc-particles. The overall uncertainty in cross sections including both systematic and statistical uncertainties, should be less than * 10 % in most cases.

3. Calculations 3.1. ELASTIC

SCATTERING

The analysis of the data depends on the availability of optical-model potentials for the triton and deuteron at the appropriate energies. We have used for the deuteron channel the potential of Schwandt and Haeberli i2) who studied deuteron elastic scattering by 40Ca in the range of deuteron energies Ed from 5-34 MeV. A slight extrapolation is made to handle the region E,, z 4 MeV. Further, we assume 41Ca and 40Ca to be represented by a common optical-model potential taken to be of the form Yd(r)

= U,(r) - Vf(x) + 4iW $

f(x’) + 21/,.,. k d”, f(x”)L

. S,

(‘1

where f(x)

= (1 +exp x)-l,

(2)

x = a;‘(r--+I+), 5’ = a,‘(r-rr,Af),

(3)

x” = osTof(r - r,,:, A+).

Here, U,-(r) is the Coulomb potential for a charged sphere of radius 1.3A” and S is the spin-one operator. The parameters are listed in table 1. We notice little change in the

340

L. J. B. GOLDFARB

geometry throughout with Ed.

the energy region;

although

et al.

there is a significant

The classical distances of closest approach corresponding to energies are M 7 fm for the deuteron and 14 fm for the tritons. the distortion in the deuteron channel to be dominant. Explicit shows that similar importance must be attached to the distortion which points again to poor spatial localization in these reactions.

TABLE

increase

the lowest This would calculation in the triton

of W

reaction suggest however channel

1

Parameters for the deuteron optical model Ed “)

V “)

5.6 5.1

112 112

4.6 4.1

113 113

rv

b,

1.05

“) In MeV.

a,

b,

b,

9

Wa)

r,

9.5 9.0

1.65 1.66

0.52 0.52

8.5 1.5

1.66 1.61

0.51 0.51

6

0.85

K.,. “)

r,.,. b,

9.0

0.9

6.0.

b,

0.6

b, In fm.

No elastic scattering data existed for the tritons in this energy region and it was decided to obtain an excitation curve for tritons at the lab angle of 160” where the nuclear distortion is expected to be significant. This proved to be inadequate for a detailed analysis; but we were fortunate to have made available to us sets of elastic scattering measurements performed by the Strasbourg group 14) between 2.5 MeV and 3.5 MeV. Our analysis, based on a parameter-search facility, points to no need to change the geometry in this energy region just as was found for deuterons. We chose a spin-independent potential for the tritons of the form

V;(r) = U,(r) - Vf(x)-

i Wf(x’),

(4)

the notation being that used for eq. (1). The ambiguities in parameterization were found to be considerable at these energies. Those that were apparent were of the types Vrl: = constant and Wrft = constant where n is close to unity. We arbitrarily fixed I’ at 150 MeV, let a, = a, = 0.65 fm and allowed rvt, rw and W to vary. This led to the parameterization labelled Z which is shown in table 2. Guided by the results of on 40Ca, we set other elastic scattering analysis I28 16*17) of various projectiles V = 150 MeV, fixed rv, av and ow to be 1.10 fm and 0.65 fm, respectively and allowed W to vary. This led to the parameterization labelled II in table 2. Also considered is the potential of Rook 15), labelled R, who studied triton elastic scattering by 40Ca between 6.4 and 7.2 MeV and the potential of Cline et al. 1“) labelled C, who considered helion scattering between 8 and 10.25 MeV.

rw and

341

40Ca(t, d0)4’Ca REACTION TABLE2 Parameters Potential I

II

R

C

“) In MeV.

I$ (lab) “) 3.5 3.0 2.5 2.0 3.5 3.0

VP)

for the triton optical model rv v

aV Y

WB)

0.65

12 12 i

rw Y

aw Y

17

150

1.39

1.58

0.65

4 23 150

2.0 2.5 i 3.5 3.0 2.5 2.0 i 3.5 3.0 2.5 2.0

150

184

1.10

1.30

1.07

0.80

:;

)

1.60

0.65

0.65

12 20 \ 17 1o

1.30

0.65

0.854

4 9.5 9.0 85

1.81

0.592

X2 37 29 5.2 20 28 9.4 40 55 19 204 13.2 13.6

715

b, In fm.

0

30

60

90 Bc.m.

120

t50

180

Fig. 1. Angular distributions for tritons of energy 2.5,3-O and 3.5 MeV scattered by 4oCa. The experimental points are those of ref. 4, and the calcuIations are based on the potential R.

The Rook potential, to the elastic scattering to express our results fits to both processes. R. The triton energies

although it provides a slightly worse fit than potentials I and II data, gave the best fit to the transfer process and we decided in terms of this potential. The potential C gave fairly inferior Fig. 1 shows a comparison with experiment using the potential are 2.5, 3.0 and 3.5 MeV.

342 3.2. THE

L. J. B. GOLDFARB “OCa(t,

dj4’Ca

GROUND

STATE

et al.

REACTION

The (t, d) reaction is commonly calculated using (d, p) DWBA computing codes based on the zero-range approximation. The differential cross section is in this case written

as

where Gcode(0) is of the form c7code(t)) = /1 dv Q(r) I22

(6)

Q(r) = xI-‘*(~)~,(r)x’+‘(y),

(7)

X!+‘(Y) and X$-)(Y) are distorted-wave eigenfunctions depicting the relative motion in the two reaction channels, $,,(r) is the single-particle wave function for the transferred neutron, having a spectroscopic factor S, in the residual nucleus and No expresses the information concerning the (tld) overlap. In studies near the Coulomb barrier, we can attempt a comparatively detailed treatment of the triton. The more precise DWBA calculation leads to the finite-range expression “) dr

OFR(@ = Is where d2 is a differential

operator

l-b,;

+h($~‘-..]

Q(r))‘,

(8)

[ acting on Q(Y) [see ref. “) for details] and X2 = 2%,(4

- Bd),

(9

the quantities B, and B,, referring respectively to the triton and deuteron system while’ KH~,,refers to the reduced mass of the neutron and deuteron. The coefficients b,, b 2, . . . vanish identically if a zero-range neutron-deuteron interaction is assumed: otherwise they form a rapidly decreasing sequence for any realistic (tld) model. In the sub-Coulomb approximation “)

!L x2

(10)

-1

’

whence

B~~~,(~) = (1 + b, + b2 + . . *> ijdra(4

I22

(11) (12)

N = N,(l+

b, + b, + . . .)‘.

(13)

The renormalization constant No is found to be very sensitive to the size of the triton “) and a commonly used value suggested by Bassel ‘) corresponds to a triton

@Ca(t,

REACTION

343

rms radius

overestimated

associated particular,

with triton wave-functions adopted to the correct using units of IO4 MeV2 * fm3 for N-values,

iv, =

by about

d0)41Ca

25 %. Our calculations

are based on values of N rms radius “) and, in

3.15, N = 4.79, b, = 0.13, b, = 0.03.

(14)

The quantity N, corresponds to the quantity N = 30; = 5.06 suggested in Bassel’s paper “). We also brought into our calculations the effects of nuclear distortion by relaxing the sub-Coulomb condition for the coefficient of b, in eq. (8). Thus “)

(15) where U, denotes the single-particle potential characterizing the binding of the neutron in the exit channel. We further utilize approximation (10) for the coefficients of b, (n > 1). This has the effect of altering the angular distribution in addition to changing the magnitude of the cross section. A calculation such as this attempts to treat the finite-range corrections beyond first-order as in eq. (15) by employing the Coulomb approximation [eq. (10)) for the higher-order terms. II

0

1

30

I

11

I

60

II

90 e c.m.

II

1

120

II

1

150

II

180

Fig. 2. Angular distributions for the 40Ca(t, d)‘Wa ground state reaction at laboratory energies of 2.0,2.5, 3.0 and 3.5 MeV. The dashed curves follow the use of eq. (12) for dc/dQ while the continuous curves show further finite-range corrections with the coefficient of b, altered as in eq. (15), but with the Coulomb approximations used for the higher terms. A spectroscopic factor of 0.77 was taken for the transition. The calculations are based on the potential R.

L. J. B. GOLDFARB

344

The results of the calculation

et al.

using the potential

R to describe the triton

scattering

are shown in fig. 2 for various triton energies. The deviation of the angular distribution from the sub-Coulomb stripping approximation (10) arising from the amendment of the coefficient of b, as in eq. (15) is found to be insignificant at all the energies. 4. Results and discussion The overall agreement with experiment is fairly good, particularly if we note that the fit is made to the angular distribution at each energy with no change in normalization. Most interesting is the fact that the results at the highest energies are crucial in removing the uncertainties concerning the bound-state potential for the neutron.

0

30

60

90

120

150

180

8 c.m. Fig.

3. Variations

in the shape

of da/dQ for the ‘?Ia(t, do)41 The triton energy is 3.5 MeV.

reaction

for selected

r,

values.

Changes in the value of r, are known to result in appreciable change to the calculated cross section at sub-Coulomb energies. This was first stressed I*) in studies of neutron transfer between heavy ions at sub-Coulomb energies. Yet, the traces of a diffraction pattern in the angular distributions at 3.5 MeV can be correlated with a specific value of r,. This is evident in fig. 3 which shows the changes in the angular distribution with variations in r,. The experimental results suggest a value of r, close to 1.25 fm if a, is taken to be 0.65 fm. Increases in a, would be associated with decreasing values of r,,. We chose however to keep to this commonly assumed value of a, and to let r,, equal 1.25 fm. The sensitivity of the shape of the angular distribution to the value of r, does not seem to be apparent in the study of (d, p) reactions on 40Ca by Seth et al. 13) at much higher energies. There it was found that increasing the value of r, by 0.05 fm

‘%a&

resulted

do)41Ca

345

REACTION

in an increase in the cross section by FZ 20 o/owith little change in the angular

distribution. The differential cross section for the stripping process shows little sensitivity to the parameterization in the deuteron channel at all the energies. Large variations in the strength of the real part of the triton optical-model interaction and in the spin-orbit t’

”

’

’ ”

”

”

”

”

’ ‘1

2.0 MeV!

0

30

90

60

120

150

180

ec.m. Fig. 4. Variations in da/dQ for the 40Ca(t , dr,)41Ca reaction for selected values of W,, using either the potential R or II to depict the triton interaction. Also shown for a triton energy of 2.0 MeV is do/dQ as calculated using only Coulomb wave functions for the entrance and exit channels.

TABLE 3 Spectroscopic

factors

for different

triton

Potential

S(lf$

Neutron U” “) 53.4 “) In MeV.

b, In fm.

I II R bound r, 9 1.25

optical-model

potentials

0.84 0.80 0.17 state parameters a” 9 0.65

V s.0. “) 5.0

336

L. J. B. GOLDFARB

et nl.

term are also of minor importance. There is a sensitivity to l+i, if less than This is particularly relevant to the situation when Et = 2 MeV. Apart from tation curve at 160“ there are no eiastic scattering data at this energy; however unlikely, even with the availability of this data, that the value of W, would

10 MeV. the exciit seems be \sell-

determined. Only the potentials R and I involve shaltow wells and, to illustrate this feature, the effect of altering ct; is shown in fig. 4 for potentials R and II. We have proposed for the R-potential a value equal to 4 MeV. A different W, value would merely result in a spectroscopic factor inconsistent with what is found at the higher energies. Table 3 lists the spectroscopic factors S, corresponding to the three triton optical potentials labelled I, II and R. The average value of S,, is 0.80+0.03, the main LHIcertainty being that associated with the ambiguities in the triton optical potential. There is also an uncertainty associated with the choice of Y,, value. Schwandt and Haeberli “) on the other hand found S, to vary with energy from 0.80 to I.1 5 for incident deuteron energies Ed from 5 to 14.3 MeV. Seth, Picard and Satchler r3) found S, to be 0.7 ri: 0.1 corresponding to Ed in the range from 11 to 12 MeV. If they had chosen r, to equal 1.25 fm instead of the values of 1.I7 and 1.20 fm assumed in their analysis, S, would have been 0.82 which is in good agreement with our result. Our analysis points only to a small dependence of the derived quantities of S, on the nature of the distorted waves. It suffers though in the need to have definite information concerning the internal state of the triton. The purpose of our study is however to go beyond establishing S,. We are most interested in confirming the Coulomb stripping theory as presented in ref. “) and further developed in ref. “). A value of S, near 0.80 is in accord with the work of Cerace and Green I’). Early views of 40Ca as a doubly closed shell nucleus are not supported by the presence of nearby states with well-pronounced rotational character. Gerace and Green modified the description of the low-lying states of 40Ca and ‘ICa by adding to the usual 2sld orbitals specific deformed states formed by raising an even number of particles to the next 2plf shell. From this, a spectroscopic factor of 0.9 was deduced for the ground state of 4’Ca. The reasonable value of S, equal to 0.801_+0.03 suggest the validity of N, and N as given in eq. (14). If, indeed, we had used Bassel’s value of “N” = 5.06, i.e., a value of N, = N = 3.15, and inserted this into the expression for do/dQ in eq. (5), the value of S,, would equal 1.2 which is hardly reasonable. Implicit in the foregoing analysis is the assumption that the reaction can be treated solely as a direct process. Such an assumption is physically fairly reasonable since the cIassica1 distances of closest approach range over values far in excess of the nuclear radius. It is also possible that the large spin change between **Ca and 4’Ca would tend to diminish compound nucleus contributions because of the relatively low penetrabilities that are involved. Some attempts to assess the importance of such contributions was made by measurement of the excitation curves for the elastic scattering and for the 40Ca(t, do)41Ca reaction at a laboratory angle of 160”. The results for the reaction are shown in fig. 5 and they show no evidence for strong compound nucleus

40Ca(t,

dc,)4’Ca

347

REACTION

effects. The 100 keV energy steps and the target thickness, which varied from 40 to 75 keV, rule out the possibility of observing narrow resonances; but the smoothness of the excitation curve does appear 40Ca(t d,)41Ca reaction proceeds The Conclusions drawn mentation, but with nuclei other (t, d) analysis of this be done with intermediate

I

I

. EXPERIMENT 0 THEORY

a*

0

cl

the

from this experiment point to the value of further experimore suited to sub-Coulomb conditions. To-date, the only kind has been in the lead region and there is much that can nuclei. I

1.0

to be consistent with our assumption that entirely by a direct reaction mechanism.

1.5

2.0

l+a

.

2.5

_

3.0

3.5

E + (LAB) MeV

Fig.

5. Excitation curve for the 40Ca(t , d0)41Ca reaction theoretical points are taken from the fits shown

at a laboratory angle of in fig. 3 using potential R.

160”. The

We wish to thank Professor D. Magnac-Valette and Mr. 0. Bing for kindly allowing us to use their recently acquired elastic scattering data of tritons on 40Ca.

References I) L. J. B. Goldfarb, Nucl. Phys. 72 (1965) 537 2) R. H. Bassel, Phys. Rev. 149 (1966) 791 3) L. J. B. Goldfarb and E. Parry, Nucl. Phys. All0 (1968)289 4) L. J. B. Goldfarb and J. A. Gonzalez, to be published 5) W. R. Hering, M. Dost and U. Lynen, Phys. Lett. 21 (1966) 695 6) A. F. Jeans, K. W. Jones and W. Darcey, Phys. Lett. 27B (1968) 431 7) G. J. Igo, P. D. Barnes, E. R. Flynn and D. D. Armstrong, Phys. Rev. 177 (1969) 1831; A. R. Barnett, P. J. A. Buttle, L. J. B. Goldfarb and W. R. Phillips, Nucl. Phys. Al76 (1971) 321 8) L. L. Lee, Jr., J. P. Schiffer, B. Zeidman, G. R. Satchler, R. M. Drisko and R. H. Bassel, Phys. Rev. 136 (1964) B971

348 9) 10) 11) 12) 13) 14) 15) 16) 17)

L. J. B. GOLDFARB

er al.

S. A. Hjorth, J. X. Saladin and G. R. Satchler, Phys. Rev. 138 (1965) B1425 H. G. Leighton, G. Roy, D. P. Curd and T. B. Grandy, Nucl. Phys. A109 (1968) 218 C. A. Pearson and D. Zissermann, Nucl. Phys. Al54 (1970) 23 P. Schwandt and W. Haeberli, Nucl. Phys. A123 (1969) 401 K. K. Seth, J. Picard and G. R. Satchler, Nucl. Phys. A140 (1970) 577 D. Magnac-Valette and 0. Bing, to be published J. R. Rook, Nucl. Phys. 61 (1965) 219 D. Cline, W. Parker Alford and L. M. Blau, Nucl. Phys. 73 (1965) 33 E. F. Gibson, B. W. Ridley, J. J. Kraushaar, M. E. Rickey and R. H. Bassel, Phys. Rev. 155 (1967) 1194 and 1208 18) L. J. B. Goldfarb and J. W. Steed, Nucl. Phys. All6 (1968) 321 19) W. J. Gerace and A. M. Green, Nucl. Phys. A93 (1967) 110